Method and device for performing graph-based transform using generalized graph parameter

ABSTRACT

The present invention provides a method for decoding a video signal using a graph-based transform including receiving a generalized graph signal including a graph parameter set; obtaining a graph-based transform kernel of a transform unit based on the graph parameter set and a predetermined penalty function; and decoding the transform unit using the graph-based transform kernel.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is the National Stage filing under 35 U.S.C. 371 ofInternational Application No. PCT/KR2015/012218, filed on Nov. 13, 2015,which claims the benefit of U.S. Provisional Applications No.62/079,566, filed on Nov. 14, 2014, the contents of which are all herebyincorporated by reference herein in their entirety.

TECHNICAL FIELD

The present invention relates to a graph-based signal processing methodand apparatus, and more particularly, to a method and apparatus forprocessing a signal for performing a graph-based prediction using anoptimization function.

BACKGROUND ART

Most of the traditional discrete-time signal processing techniques haveevolved directly from processing and filtering analog signals, and thushave been constrained by several common assumptions, like sampling andprocessing only regularly organized data. The field of video compressionis based on basically the same assumptions, but only generalized tomulti-dimensional signals.

A graph is a useful type of data representation for describing ageometrical structure of data in various application fields. Signalprocessing based on graph representations can generalize concepts likesampling, filtering, Fourier transforms, etc., using graphs where eachsignal sample represents a vertex, and signal relationships arerepresented by graph edges with positive weights. This disconnects thesignal from its acquisition process, so that properties like samplingrate and sequences can be replaced by the properties of graph.Accordingly, more efficient signal processing method based on graph isrequired in many application fields as well as in the field of videocompression.

DISCLOSURE Technical Problem

Graph-based signal processing is a new technical field that generalizesvarious techniques and models of signal processing, and has alreadyshown a promising result in many application fields. However, in thecase that the graph-based signal processing technique is applied whencompressing signals, in order to produce better signal transformation orprediction, it is required for both of an encoder and a decoder to usethe same graph (e.g., a vertex, an edge and an edge weight). Althoughmore improved signal compression may be attained using more complex andadaptive graph structure, the overhead for coding the information of thegraph becomes greater relatively. Accordingly, there is a problem thatthe overhead becomes greater than the gain provided by it. Therefore,the present invention is to solve such a problem.

In addition, the present invention is to provide a new method forcalculating a graph-based transform using generalization of theconventional spectral decomposition. In addition, the present inventionis to better control transform properties through such a generalization,and to be applicable to various applications.

Furthermore, the present invention is to propose a method for detectinga vertex and an edge in a graph and for encoding or decoding a residualsignal.

In addition, the present invention develops a graph-based tool forbetter coding set of prediction residual values.

Technical Solution

The present invention provides a method for generalizing the definitionof graph parameters.

In addition, the present invention provides a method for obtaining agraph signal using at least one of a vertex parameter set and an edgeparameter set.

In addition, the present invention provides a method for obtaining anoptimized transform kernel based on at least one of a penalty functionand a constraint function.

In addition, the present invention provides a method for deriving aneigenvalue function by applying a penalty function predefined for agraph signal.

In addition, the present invention provides a method for obtaining agraph-based transform kernel that optimizes an eigenvalue function.

In addition, the present invention proposes a method for detecting avertex and an edge in a graph and for encoding or decoding a residualsignal.

Technical Effects

The graph-based signal modeling to which the present invention isapplied may be a powerful tool. Particularly, the present inventionprovides a new method for calculating a graph-based transform usinggeneralization of the conventional spectral decomposition, therebyavoiding the sharp discontinuity problem of vectors for defining thegraph-based transform.

In addition, the present invention may better control the transformproperties through the generalization of the conventional spectraldecomposition, and may be applicable to various applications.

In addition, the present invention may improve the compressionefficiency by adaptively using the statistical properties of a signal indifferent parts of a video sequence.

In addition, the present invention may avoid an excessive overhead of abit rate required to encode a graph signal through the generalization ofthe conventional spectral decomposition.

DESCRIPTION OF DRAWINGS

FIG. 1 shows a schematic block diagram of an encoder for encoding avideo signal, in accordance with one embodiment of the presentinvention.

FIG. 2 shows a schematic block diagram of a decoder for decoding a videosignal, in accordance with one embodiment of the present invention.

FIG. 3 shows examples of graphs used for modeling statisticalrelationships in 8×8 block within a video frame according to anembodiment to which the present invention is applied.

FIG. 4 shows a graph of two shapes representing weights distribution asan embodiment to which the present invention is applied.

FIG. 5 is a diagram for describing a procedure of obtaining agraph-based transform matrix based on 1-dimensional graph and2-dimensional graph as an embodiment to which the present invention isapplied.

FIG. 6 illustrates a schematic block diagram of an encoder forprocessing a graph-based signal as an embodiment to which the presentinvention is applied.

FIG. 7 illustrates a schematic block diagram of a decoder that processesa graph-based signal as an embodiment to which the present invention isapplied.

FIG. 8 illustrates an inner block diagram of a graph-based transformunit as an embodiment to which the present invention is applied.

FIG. 9 is a flowchart for describing a procedure of calculating anoptimized transform matrix based on a generalized graph parameter and apenalty function, as an embodiment to which the present invention isapplied.

FIG. 10 is a flowchart for describing a procedure of obtaining agraph-based transform kernel using a generalized parameter set, as anembodiment to which the present invention is applied.

BEST MODE FOR INVENTION

The present invention provides a method for decoding a video signalusing a graph-based transform including receiving a generalized graphsignal including a graph parameter set; obtaining a graph-basedtransform kernel of a transform unit based on the graph parameter setand a predetermined penalty function; and decoding the transform unitusing the graph-based transform kernel.

In addition, in the present invention, the graph parameter set includesat least one of a vertex parameter set represented as V-dimensionalvector and an edge parameter set represented as V×V matrix.

In addition, in the present invention, the predetermined penaltyfunction is generated based on the generalized graph signal.

In addition, in the present invention, the graph-based transform kernelis obtained by an optimization function based on the graph parameter setand the predetermined penalty function.

In addition, in the present invention, wherein the optimization functionis comprised of a summation of a first penalty function component for avertex parameter set and a second penalty function component for an edgeparameter set, and wherein the graph-based transform kernel indicates avalue in which the optimization function is a minimum.

In addition, the present invention provides a method for performing agraph-based transform based on a generalized graph signal includingdetermining a graph parameter including at least one of a vertexparameter set and an edge parameter set; generating a generalized graphsignal based on the graph parameter; generating at least one of apenalty function and a constraint function based on the graph parameter;generating an optimization function based on at least one of the penaltyfunction and the constraint function and the generalized graph signal;obtaining an optimal graph-based transform kernel in which theoptimization function is a minimum; and performing a transform for atransform unit using the optimal graph-based transform kernel.

In addition, the present invention provides an apparatus for decoding avideo signal using a graph-based transform an entropy decoding unitconfigured to receive a generalized graph signal including a graphparameter set; and an inverse transform unit configured to obtain agraph-based transform kernel of a transform unit based on the graphparameter set and a predetermined penalty function, and to decode thetransform unit using the graph-based transform kernel.

In addition, the present invention provides an apparatus for performinga graph-based transform using a generalized graph signal a graph signalgenerating unit configured to determine a graph parameter including atleast one of a vertex parameter set and an edge parameter set, and togenerate a generalized graph signal based on the graph parameter; atransform matrix obtaining unit configured to generate at least one of apenalty function and a constraint function based on the graph parameter,to generate an optimization function based on at least one of thepenalty function and the constraint function and the generalized graphsignal, and to obtain an optimal graph-based transform kernel in whichthe optimization function is a minimum; and a transform performing unitconfigured to perform a transform for a transform unit using the optimalgraph-based transform kernel.

MODE FOR INVENTION

Hereinafter, exemplary elements and operations in accordance withembodiments of the present invention are described with reference to theaccompanying drawings, however, it is to be noted that the elements andoperations of the present invention described with reference to thedrawings are provided as only embodiments and the technical spirit andkernel configuration and operation of the present invention are notlimited thereto.

Furthermore, terms used in this specification are common terms that areFurthermore, terms used in this specification are common terms that arenow widely used, but in special cases, terms randomly selected by theapplicant are used. In such a case, the meaning of a corresponding termis clearly described in the detailed description of a correspondingpart. Accordingly, it is to be noted that the present invention shouldnot be construed as being based on only the name of a term used in acorresponding description of this specification and that the presentinvention should be construed by checking even the meaning of acorresponding term.

In addition, embodiments proposed in this specification is directed tovideo signal processing, but the present invention should not beconstrued as being based on only video signal processing, and may beapplicable to a method of processing general graph-based signal.

Furthermore, terms used in this specification are common terms selectedto describe the invention, but may be replaced with other terms for moreappropriate analysis if such terms having similar meanings are present.For example, a signal, data, a sample, a picture, a frame, and a blockmay be properly replaced and interpreted in each coding process.

By applying a linear transform that adaptively modifies the statisticalproperties of a signal in different parts of a video sequence,compression efficiency may be improved. General statistical methods havebeen tried such an object, but they bring a restricted result. Thepresent invention introduces a graph-based signal processing techniqueas a more efficient method for modeling the video statistical propertiesfor video compression.

In order to simplify mathematical analysis and to use the result knownfrom a graph theory, most of applications developed for the graph-basedsignal processing uses an undirected graph without self-loop (i.e.,there is no edge that connects nodes in itself.), and models withnon-negative edge only in each graph edge.

Such an approach may be successfully applied for signaling an image ofwell defined discontinuity, sharp edge or a depth image. The graphscorresponding to N² pixel blocks in an image and video applicationrequire transmission overhead for 2N² or 4N² non-negative edge weights,generally. After a graph is defined, the orthogonal transform for codingor prediction may be induced by calculating spectral decomposition of agraph Laplacian matrix. For example, through the spectral decomposition,an eigenvector and an eigenvalue may be obtained.

The present invention provides a new method for modifying the procedureof calculating a graph-based transform using new generalization of theconventional spectral decomposition. Here, the transform obtained from agraph signal may be defined as Graph-Based Transform (hereinafter, GBT).For example, when the relation information between pixels constructing aTU is represented in a graph, the transform obtained from the graph maybe referred to as GBT.

The general form of the spectral decomposition to which the presentinvention is applied may be obtained based on an additional set of graphedge parameters that have desired properties and graph vertexparameters. Through such an embodiment of the present invention, thetransform properties may be well controlled, and the problem of sharpdiscontinuities of the vectors defining transform may be avoided.Hereinafter, the embodiments to which the present invention will bedescribed in detail.

FIG. 1 shows a schematic block diagram of an encoder for encoding avideo signal, in accordance with one embodiment of the presentinvention.

Referring to FIG. 1, an encoder 100 may include an image segmentationunit 110, a transform unit 120, a quantization unit 130, an inversequantization unit 140, an inverse transform unit 150, a filtering unit160, a DPB (Decoded Picture Buffer) 170, an inter-prediction unit 180,an intra-prediction unit 185 and an entropy-encoding unit 190.

The image segmentation unit 110 may divide an input image (or, apicture, a frame) input to the encoder 100 into one or more processunits. For example, the process unit may be a coding tree unit (CTU), acoding unit (CU), a prediction unit (PU), or a transform unit (TU).

However, the terms are used only for convenience of illustration of thepresent disclosure. The present invention is not limited to thedefinitions of the terms. In this specification, for convenience ofillustration, the term “coding unit” is employed as a unit used in aprocess of encoding or decoding a video signal. However, the presentinvention is not limited thereto. Another process unit may beappropriately selected based on contents of the present disclosure.

The encoder 100 may generate a residual signal by subtracting aprediction signal output from the inter-prediction unit 180 or intraprediction unit 185 from the input image signal. The generated residualsignal may be transmitted to the transform unit 120.

The transform unit 120 may apply a transform technique to the residualsignal to produce a transform coefficient. The transform process may beapplied to a pixel block having the same size of a square, or to a blockof a variable size other than a square.

In an embodiment of the present invention, the transform unit 120 mayobtain a graph signal using a generalized graph parameter.

In another embodiment of the present invention, the transform unit 120may obtain a graph signal using at least one of a vertex parameter setand an edge parameter set, and may derive an eigenvalue function byapplying a predefined penalty function to the graph signal.

In another embodiment of the present invention, the transform unit 120may obtain an optimized transform kernel based on at least one of apenalty function and a constraint function. In this case, the optimizedtransform kernel may be a value that optimizes the eigenvalue function.

The quantization unit 130 may quantize the transform coefficient andtransmits the quantized coefficient to the entropy-encoding unit 190.The entropy-encoding unit 190 may entropy-code the quantized signal andthen output the entropy-coded signal as bitstreams.

The quantized signal output from the quantization unit 130 may be usedto generate a prediction signal. For example, the quantized signal maybe subjected to an inverse quantization and an inverse transform via theinverse quantization unit 140 and the inverse transform unit 150 in theloop respectively to reconstruct a residual signal. The reconstructedresidual signal may be added to the prediction signal output from theinter-prediction unit 180 or intra-prediction unit 185 to generate areconstructed signal.

On the other hand, in the compression process, adjacent blocks may bequantized by different quantization parameters, so that deterioration ofthe block boundary may occur. This phenomenon is called blockingartifacts. This is one of important factors for evaluating imagequality. A filtering process may be performed to reduce suchdeterioration. Using the filtering process, the blocking deteriorationmay be eliminated, and, at the same time, an error of a current picturemay be reduced, thereby improving the image quality.

The filtering unit 160 may apply filtering to the reconstructed signaland then outputs the filtered reconstructed signal to a reproducingdevice or the decoded picture buffer 170. The filtered signaltransmitted to the decoded picture buffer 170 may be used as a referencepicture in the inter-prediction unit 180. In this way, using thefiltered picture as the reference picture in the inter-pictureprediction mode, not only the picture quality but also the codingefficiency may be improved.

The decoded picture buffer 170 may store the filtered picture for use asthe reference picture in the inter-prediction unit 180.

The inter-prediction unit 180 may perform temporal prediction and/orspatial prediction with reference to the reconstructed picture to removetemporal redundancy and/or spatial redundancy. In this case, thereference picture used for the prediction may be a transformed signalobtained via the quantization and inverse quantization on a block basisin the previous encoding/decoding. Thus, this may result in blockingartifacts or ringing artifacts.

Accordingly, in order to solve the performance degradation due to thediscontinuity or quantization of the signal, the inter-prediction unit180 may interpolate signals between pixels on a subpixel basis using alow-pass filter. In this case, the subpixel may mean a virtual pixelgenerated by applying an interpolation filter. An integer pixel means anactual pixel existing in the reconstructed picture. The interpolationmethod may include linear interpolation, bi-linear interpolation andWiener filter, etc.

The interpolation filter may be applied to the reconstructed picture toimprove the accuracy of the prediction. For example, theinter-prediction unit 180 may apply the interpolation filter to integerpixels to generate interpolated pixels. The inter-prediction unit 180may perform prediction using an interpolated block composed of theinterpolated pixels as a prediction block.

The intra-prediction unit 185 may predict a current block by referringto samples in the vicinity of a block to be encoded currently. Theintra-prediction unit 185 may perform a following procedure to performintra prediction. First, the intra-prediction unit 185 may preparereference samples needed to generate a prediction signal. Then, theintra-prediction unit 185 may generate the prediction signal using theprepared reference samples. Thereafter, the intra-prediction unit 185may encode a prediction mode. At this time, reference samples may beprepared through reference sample padding and/or reference samplefiltering. Since the reference samples have undergone the prediction andreconstruction process, a quantization error may exist. Therefore, inorder to reduce such errors, a reference sample filtering process may beperformed for each prediction mode used for intra-prediction

The prediction signal generated via the inter-prediction unit 180 or theintra-prediction unit 185 may be used to generate the reconstructedsignal or used to generate the residual signal.

FIG. 2 shows a schematic block diagram of a decoder for decoding a videosignal, in accordance with one embodiment of the present invention.

Referring to FIG. 2, a decoder 200 may include an entropy-decoding unit210, an inverse quantization unit 220, an inverse transform unit 230, afiltering unit 240, a decoded picture buffer (DPB) 250, aninter-prediction unit 260 and an intra-prediction unit 265.

A reconstructed video signal output from the decoder 200 may bereproduced using a reproducing device.

The decoder 200 may receive the signal output from the encoder as shownin FIG. 1. The received signal may be entropy-decoded via theentropy-decoding unit 210.

The inverse quantization unit 220 may obtain a transform coefficientfrom the entropy-decoded signal using quantization step sizeinformation. In this case, the obtained transform coefficient may beassociated with the operations of the transform unit 120 as describedabove with reference to FIG. 1.

The inverse transform unit 230 may inverse-transform the transformcoefficient to obtain a residual signal.

A reconstructed signal may be generated by adding the obtained residualsignal to the prediction signal output from the inter-prediction unit260 or the intra-prediction unit 265.

The filtering unit 240 may apply filtering to the reconstructed signaland may output the filtered reconstructed signal to the reproducingdevice or the decoded picture buffer unit 250. The filtered signaltransmitted to the decoded picture buffer unit 250 may be used as areference picture in the inter-prediction unit 260.

Herein, detailed descriptions for the filtering unit 160, theinter-prediction unit 180 and the intra-prediction unit 185 of theencoder 100 may be equally applied to the filtering unit 240, theinter-prediction unit 260 and the intra-prediction unit 265 of thedecoder 200 respectively.

FIG. 3 shows examples of graphs used for modeling statisticalrelationships in 8×8 block within a video frame according to anembodiment to which the present invention is applied.

The discrete-time signal processing technique has been developed fromdirectly processing and filtering an analogue signal, and accordingly,has been restricted by a few common assumptions such as sampling andprocessing regularly organized data only.

Basically, the video compression field is based on the same assumption,but has been generalized for a multi-dimensional signal. The signalprocessing based on a graph representation generalizes the concepts suchas sampling, filtering and Fourier transform, uses the graph thatrepresents a vertex by each signal sample, and is started from theconventional approach in which signal relationships are represented bygraph edges with positive weights. This completely isolates a signalfrom its acquisition process, and accordingly, the properties such assampling rate and sequence are completely replaced by the properties ofa graph. Accordingly, the graph representation may be defined by a fewspecific graph models.

In order to represent an empirical connection between data values, thepresent invention has an undirected simple graph and an undirected edgeonly, normally. Here, the undirected simple graph may mean a graphwithout self-loop or multiple edges.

When the undirected simple graph that has a weight allocated for eachedge is referred to as G, the undirected simple graph G may be describedwith triplet as represented in Equation 1.

={

,ε,W}  [Equation 1]

Here, V represents V numbers of graph vertex set, ε represents a graphedge set, and W represents a weight represented as V×V matrix. Here,weight W may be represented as Equation 2 below.W _(i,j) =W _(j,i)≥0  [Equation 2]

W_(i,j) represents a weight of edge (i, j), and W_(j,i) represents aweight of edge (j, i). When there is no edge connecting vertex (i, j),W_(i,j)=0. For example, in the case of assuming that there is noself-loop, W_(i,i)=0, always.

The representation is partially overlapped for a special case of theundirected simple graphs that have an edge weight. This is becausematrix W includes all types of information of the graph. Accordingly, inthe present invention, hereinafter, a graph is represented as G(W).

Meanwhile, referring to FIG. 3, the present invention provides twoembodiments of graph types that may be used for processing 8×8 pixelblocks in an image or a video. Each pixel is in relation to a graphvertex, and the pixel value becomes the value of the graph vertex.

A graph edge may mean a line connecting graph vertexes. The graph edgeis used for representing a certain type of statistical dependency withina signal, and in this case, a positive weigh may represent thesharpness. For example, each vertex may be connected to all of othervertexes, and weight of 0 may be allocated to an edge that connectsvertexes not coupled with each other or weakly coupled. However, forsimplifying the representation, the edge having the weight of 0 may becompletely removed.

As another embodiment of the present invention, the edges connectinggraph vertexes may be preconfigured depending on a signal property. Forexample, the vertexes may be arranged on 1-dimensional array for anaudio signal, on 2-dimensional array for an image, and on 3-dimensionalarray for a video frame. In this case, for the 3-dimensional array, atime axis may be the third dimension. For example, in the graph shown inFIG. 3(a), a graph edge may be defined such that each vertex isconnected to the nearest 4 adjacent vertexes. However, a block edge maybe differently treated. In addition, in the graph shown in FIG. 3(b), itmay be defined that each vertex is connected to the nearest 8 adjacentvertexes.

FIG. 4 shows a graph of two shapes representing weights distribution asan embodiment to which the present invention is applied.

The vertex value of a graph is an independent variable based on a signalmeasurement (normally, modeled as an arbitrary variable), but it isrequired to select an edge weight in accordance with the property of apart of signal. FIG. 4 shows two exemplary graphs that represent theedge weights of different lines for a graph edge. For example, the boldlines may represent the weight of w=1, and the fine lines may representthe weight of w=0.2.

The graph shown in FIG. 4(a) represents the case of having “weak link”along a straight line, and represents the case of having two edgeweights only. Here, the “weak link” means having relatively small edgeweight.

This is commonly used in a graph-based image processing actually, andsuch a construction may represent a difference between an edge in animage and a pixel statistics between different sides.

FIG. 4(b) represents a distribution of an edge weight that coversirregular area. The present invention is to provide a method forprocessing a signal using such a distribution graph of an edge weight.

FIG. 5 is a diagram for describing a procedure of obtaining agraph-based transform matrix based on 1-dimensional graph and2-dimensional graph as an embodiment to which the present invention isapplied.

As an embodiment of the present invention, the graph type that may beused for processing a pixel block in an image may be described usingFIG. 5. For example, FIG. 5(a) shows 1-dimensional graph thatcorresponds to each line in the pixel block, and FIG. 5(b) shows2-dimensional graph that corresponds to the pixel block.

A graph vertex is in relation to each pixel of the pixel block, and avalue of the graph vertex may be represented as a pixel value. And, agraph edge may mean a line connecting the graph vertexes. The graph edgeis used for representing a certain type of statistical dependency in asignal, and the value representing its sharpness may be referred to asan edge weight.

For example, FIG. 5(a) shows a 1-dimensional graph, 0, 1, 2 and 3represents the position of each vertex, and w₀, w₁ and w₂ represent theedge weight between vertexes. FIG. 5(b) shows a 2-dimensional graph, anda_(ij) (i=0, 1, 2, 3, j=0, 1, 2) and b_(ki) (k=0, 1, 2, l=0, 1, 2, 3)represent the edge weight between vertexes.

Each vertex may be connected to all of other vertexes, and weight of 0may be allocated to an edge that connects vertexes not coupled with eachother or weakly coupled. However, for simplifying the representation,the edge having the weight of 0 may be completely removed.

The relationship information between pixels may be represented aswhether there is an edge between pixels and an edge weight when eachpixel is mapped to a vertex of a graph.

In this case, the GBT may be obtained through the following procedures.For example, an encoder or a decoder may obtain graph information from atarget block of a video signal. From the obtained graph information,Laplacian matrix L may be obtained as represented in Equation 3 below.L=D−A  [Equation 3]

In Equation 3 above, D represents a degree matrix. For example, thedegree matrix may mean a diagonal matrix including the information of adegree of each vertex. A represents an adjacency matrix that representsthe interconnection (edge) with an adjacent pixel by a weight.

And, with respect to the Laplacian matrix L, a GBT kernel may beobtained by performing an eigen decomposition as represented in Equation4 below.L=U

U ^(T)  [Equation 4]

In Equation 4 above, L means a Laplacian matrix L, U means an eigenmatrix, and U^(T) means a transposed matrix of U. In Equation 4, theeigen matrix U may provide a graph-based Fourier transform specializedfor a signal suitable for the corresponding model. For example, theeigen matrix U that satisfies Equation 4 may mean a GBT kernel.

FIG. 6 illustrates a schematic block diagram of an encoder forprocessing a graph-based signal as an embodiment to which the presentinvention is applied.

The Fourier transform or the discrete Fourier transform is a basic toolfor signal processing. There is also a graph Fourier transform.Generally, such transforms are identically applied for a few specialgraphs, but may provide a much wider generalized shape that may beapplied to various applications in the present invention. In theembodiments to which the present invention is applied, a graph mayprovide insight for a signal property. In order to define a Fouriertransform of a graph, the present invention may be represented by adegree matrix that corresponds to G (W). Here, the degree matrix is adiagonal matrix including the information of a degree of each vertex,and may be defined as Equation 5 below. For example, the degree may meanthe number of a side connected to a vertex.

$\begin{matrix}{D_{i,j} = \left\{ \begin{matrix}{\sum\limits_{k = 1}^{V}W_{i,k,}} & {i = j} & \; \\\; & \; & {i,{j = 1},2,\ldots\mspace{14mu},{V.}} \\0_{i} & {{i \neq j},} & \;\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

Graph Laplacian matrix L=D−W, and accordingly, graph Laplacian matrixL_(i,j) is as represented as Equation 6 below.

$\begin{matrix}{L_{i,j} = \left\{ \begin{matrix}{\sum\limits_{k = 1}^{V}W_{i,k,}} & {i = j} & \; \\\; & \; & {i,{j = 1},2,\ldots\mspace{14mu},{V.}} \\{- W_{i,j,}} & {{i \neq j},} & \;\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

In this case, when matrix T is the graph Fourier transform, matrix T isas represented as Equation 7 below.T(W)=U ^(T)  [Equation 7]

Herein, U represents an eigen matrix that diagonalizes L, and L is asrepresented as Equation 8 below.L=UΛU ⁻¹ =UΛU ^(T)  [Equation 8]

And, an orthogonal matrix satisfies Equation 9 below.U ^(T) =U ⁻¹  [Equation 9]

Based on the definition above, the columns of eigen matrix U includeseigenvectors of L, and eigenvalues of L may be represented as Equation10 below.Λ=diag(λ),  [Equation 10]

Generally, eigenvectors are not defined as a specific shape. However,according to an object of the present invention, since L is symmetric,it should be considered that all eigenvectors are real values, and atleast one of decomposition may be existed. This may be applied to anymatrix that satisfies Equation 8.

In a few applications to which the present invention is applied, atransform matrix may be obtained from the spectral decomposition of anormalized Laplacian matrix as represented in Equation 11 below.L=D ^(−1/2) LD ^(−1/2)  [Equation 11]

In order to define Fourier transform, graph Laplacian matrix L may berepresented as Equation 12 below, and for the eigenvectors of graphLaplacian matrix L, the present invention may obtain Equation 13 below.

$\begin{matrix}{{{h^{T}L\mspace{11mu} h} = {\sum\limits_{i = 1}^{V}{\sum\limits_{j = 1}^{V}{W_{i,j}\left( {h_{i} - h_{j}} \right)}^{2}}}},} & \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack \\{{\lambda_{k} = {{u_{k}^{T}L\mspace{11mu} u_{k}} = {\sum\limits_{i = 1}^{V}{\sum\limits_{j = 1}^{V}{W_{i,j}\left( {U_{i,k} - U_{j,k}} \right)}^{2}}}}},\mspace{20mu}{k = 1},2,\ldots\mspace{14mu},{V.}} & \left\lbrack {{Equation}\mspace{14mu} 13} \right\rbrack\end{matrix}$

Here, λ_(k) represents an eigenvalue of graph Laplacian matrix L.

In the present invention, in the case that eigenvalues may bedistinguished in an ascending order, the Equations above may mean thatthe corresponding eigenvectors define the graph vertex values thatincrease the sum in which a weight is applied for a square difference.In a general signal processing, such a property may be satisfied by asine wave of an increasing frequency. In addition, such a property maybe used for generalizing a frequency concept in the graph-basedtransform that corresponds to Laplacian eigenvalues.

The spectral decomposition may be efficiently calculated by a few otheralgorisms, for example, Jacobi, Givens, Householder method, and so on.However, in the graph-based signal processing, the present invention isto consider that eigenvectors may be calculated as {U₁, U₂, . . . ,U_(V)} consequently using Rayleigh quotient in Equation 15 below thatrepresents an optimization function.

$\begin{matrix}{{R\left( {M,x} \right)}:=\frac{x^{*}{Mx}}{x^{*}x}} & \left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack \\{{u_{k} = {{\underset{b}{\arg\mspace{14mu}\min}\left\{ \frac{h^{T}\mspace{11mu} L\mspace{14mu} h}{h^{T}\mspace{11mu} h} \right\}\mspace{14mu}{s.t.\mspace{14mu} u_{i}^{T}}h} = 0}},{i = 1},2,\ldots\mspace{14mu},{k - 1.}} & \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack\end{matrix}$

Herein, s.t. (subject to) and followings represent that Equation 15 isunder the condition of a set of constraints for the optimizationproblem.

In addition, according to the present invention, normalized eigenvaluesmay be obtained based on Equation 16 below.

$\begin{matrix}{{u_{k} = {{\underset{h}{\arg\mspace{14mu}\min}\left\{ {h^{T}\mspace{11mu} L\mspace{14mu} h} \right\}\mspace{14mu}{s.t.\mspace{11mu} h^{T}}h} = 1}},{{u_{i}^{T}h} = 0},{i = 1},2,\ldots\mspace{14mu},{k - 1.}} & \left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$

According to the present invention, an alternative form such as Equation17 below may be obtained using Equation 12.

$\begin{matrix}{{u_{k} = {{\underset{h}{\arg\mspace{14mu}\min}\left\{ {\sum\limits_{i = 1}^{V}{\sum\limits_{j = 1}^{V}{W_{i,j}\left( {h_{i} - h_{j}} \right)}^{2}}} \right\}\mspace{14mu}{s.t.\mspace{14mu} h^{T}}h} = 1}},{{u_{i}^{T}h} = 0},{i = 1},2,\ldots\mspace{14mu},{k - 1.}} & \left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack\end{matrix}$

Major difficulty in the formulation of the optimization function asrepresented in Equation 17 above is that it is in relation to well knowntheory and the solving method, but very restrictive. When considering aproblem of finding an optimized transform for a signal, it may beobtained by changing a non-negative weight of a graph. For example, whengraph modeling is coupled with an edge detection of a signal processing,the present invention may have a great deviation according to welldefined edges. However, in the case that an edge is blurred or theposition is unable to be accurately determined, more fine control isrequired.

In designing graph transform to which the present invention is applied,it is not required to use the sum of the square differences of Equation12 applied by a weight. In the case of changing a general definition ofgraph parameters according to the present invention, well knownproperties are lost in an aspect, but capacity occurs afford to design atransform more suitable to a specific application in the other aspect.

Accordingly, another embodiment of the present invention provides amethod for generalizing a graph related transform calculation defined bythe sequence of Equation 17 representing an optimization function. Inthe embodiment below, a method for generalizing a transform equationwill be described with a specific example.

Firstly, in the present invention, the definition of a graph parameteris required to be generalized. ‘A’ numbers of vertex parameter set maybe defined as represented in Equation 18. In this case, ‘A’ numbers ofvertex parameter set may be represented as V-dimensional vectors.v ⁽¹⁾ ,v ⁽²⁾ , . . . ,v ^((A)),  [Equation 18]

And, ‘B’ numbers of edge parameter set may be defined as represented inEquation 19 below, and this may be represented as V×V matrix.E ⁽¹⁾ ,E ⁽²⁾ , . . . ,E ^((B)),  [Equation 19]

Based on Equation 18 and Equation 19 above, in the present invention, agraph signal may be defined as Equation 20 below.

(v ⁽¹⁾ ,v ⁽²⁾ , . . . ,v ^((A)) ;E ⁽¹⁾ ,E ⁽²⁾ , . . . ,E^((B))).  [Equation 20]

In a compression application, in order to efficiently represent graphparameters, the number of encoding bits is very important factor, but itmay not be considered in the present invention.

Next, a penalty function set may be defined as Equation 21. Herein, thepenalty function represents a sort of algorithm for solving theconstraint optimization problem.P _(k):

^(V)×

^(A×V)×

^(2×B×1′) →

,k=1,2, . . . ,V,  [Equation 21]

Herein, when there is a certain constraint function in which variablevector x should be satisfied, x satisfying all of the constraintfunctions is called feasible, and in this case, a set of the feasiblepoints is referred to be as a feasible region. In the case of theoptimization problem without a constraint function, the feasible regionbecomes the whole of

^(V)×

^(A×V)×

^(2×B×1′).

Penalty function P_(k) to which the present invention is applied may berepresented as Equation 22 below.P _(k)(h;v ⁽¹⁾ ,v ⁽²⁾ , . . . ,v ^((A)) ;E ⁽¹⁾ ,E ⁽²⁾ , . . . ,E^((B))).  [Equation 22]

In addition, as represented in Equation 23, it may be defined V numbersof vector function set including the constraint function of C₁, C₂, . .. , C_(V) dimension.s _(k):

^(V×k)×

^(A×V)→

^(C) ^(k) ,k=1,2, . . . ,V,  [Equation 23]

Using the new definition of Equations 18 to 23 above, the optimizationfunction for obtaining an optimized transform matrix may be defined asEquation 24 below. Based on the optimization function of Equation 24, anoptimized transform kernel may be obtained.

$\begin{matrix}{{u_{k} = {{\underset{h}{\arg\mspace{11mu}\min}\left\{ {P_{k}\left( {{h;v^{(1)}},v^{(2)},\ldots\mspace{14mu},{v^{(A)};E^{(1)}},E^{(2)},\ldots\mspace{14mu},E^{(B)}} \right)} \right\}\mspace{14mu}{s.t.\mspace{14mu} h^{T}}h} = 1}},{{u_{i}^{T}h} = 0},{i = 1},2,\ldots\mspace{14mu},{k - 1},{{s_{k}\left( {u_{1},u_{2},\ldots\mspace{14mu},u_{k - 1},{h;v^{(1)}},v^{(2)},\ldots\mspace{14mu},v^{(A)}} \right)} \geq 0.}} & \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack\end{matrix}$

Herein, U_(k) may represent an optimized value that optimizes a targetfunction P_(k)( ) for example, may mean an optimized graph transformkernel applied to the present invention. And, “s.t.” is an acronym of“subject to”, and represents that it follows a constraint formula forthe optimization function. The column of the optimized graph transformkernel U_(k) may be sequentially calculated for k=1, 2, . . . , V.

Although Equation 24 to which the present invention is applied isrepresented to cover most of general cases, as shown in the followingembodiment, a constraint function is not necessarily used in a practicalapplication. And, the same penalty function may be repeatedly used.

The penalty functions may be defined to be calculated only for a vertexvalue. And, the parameters in the same dimension may be calculated usingthe penalty function as represented in Equation 25, and for an edgevalue difference, the penalty function as represented in Equation 26 maybe used.P _(i) ^((u)):

×

^(A) →

,i=1,2, . . . ,V,  [Equation 25]P _(i,j) ^((c)):

×

^(B) →

,i,j=1,2, . . . |,V,  [Equation 26]

Based on Equation 25 and Equation 26, an optimization function in theshape of Equation 27 below may be derived.

$\begin{matrix}{u_{k} = {\underset{h}{\arg\mspace{11mu}\min}\left\{ {{{{\sum\limits_{i = 1}^{V}{P_{i}^{(e)}\left( {h_{i},v_{i}^{(1)},v_{i}^{(2)},\ldots\mspace{14mu},v_{i}^{(A)}} \right)}} + {\left. \quad{+ {\sum\limits_{i = 1}^{V}{\sum\limits_{j = 1}^{V}{P_{i,j}^{(e)}\left( {{h_{i} - h_{j}},e_{i,j}^{(1)},\ldots\mspace{14mu},e_{i,j}^{(B)}} \right)}}}} \right\}\mspace{14mu}{s.t.\mspace{14mu} h^{T}}h}} = 1},{{u_{i}^{T}h} = 0},{i = 1},2,\ldots\mspace{14mu},{k - 1.}} \right.}} & \left\lbrack {{Equation}\mspace{14mu} 27} \right\rbrack\end{matrix}$

A simple generalization of Equation 27 above including a vertex penaltyfunction and exponents α and β may be a special case as represented inEquation 28 that corresponds to A=B=1.

$\begin{matrix}{u_{k} = {{\underset{h}{\arg\mspace{14mu}\min}\left\{ {{\sum\limits_{i = 1}^{V}{v_{i}{h_{i}}^{o}}} + {\sum\limits_{i = 1}^{V}{\sum\limits_{j = 1}^{V}{e_{i,j}{{h_{i} - h_{j}}}^{\beta}}}}} \right\}\mspace{14mu}{s.t.\mspace{14mu} h^{T}}h} = {\quad{1,{{u_{i}^{T}h} = 0},{i = 1},2,\ldots\mspace{14mu},{k - 1.}}}}} & \left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack\end{matrix}$

Meanwhile, referring to FIG. 6, an encoder 600 to which the presentinvention is applied includes a graph-based transform unit 610, aquantization unit 620, an inverse quantization unit 630, an inversetransform unit 640, a buffer 650, a prediction unit 660 and an entropyencoding unit 670.

The encoder 600 receives a video signal, and generates a predictionerror by subtracting a predicted signal outputted from the predictionunit 660 from the video signal. The generated prediction error istransmitted to the graph-based transform unit 610, and the graph-basedtransform unit 610 applies a transform scheme to the prediction error,thereby generating a transform coefficient. In this case, thegraph-based transform unit 610 may calculate a graph-based transformmatrix obtained by Equation 24, Equation 27 or Equation 28 above, andmay perform a transformation using it. In addition, graph-basedtransform unit 610 may perform the embodiments described in the presentspecification.

As another embodiment to which the present invention is applied, thegraph-based transform unit 610 may select more proper transform matrixby comparing the graph-based transform matrix obtained by Equation 24,Equation 27 or Equation 28 above with the transform matrix obtained fromthe transform unit 120 of FIG. 1 above.

The quantization unit 620 transmits the quantized coefficient to theentropy encoding unit 670 by quantizing the generated transformcoefficient.

The entropy encoding unit 670 performs entropy coding for the quantizedsignal and outputs the entropy-coded signal.

The quantized signal outputted from the quantization unit 620 may beused for generating a prediction signal. For example, the inversequantization unit 630 and the inverse transform unit 640 in a loop ofthe encoder 600 may perform the inverse quantization and the inversetransformation for the quantized signal such that the quantized signalis restored with the prediction error. The restored signal may begenerated by adding the restored prediction error to the predictionsignal outputted by the prediction unit 660.

The buffer 650 stores the restored signal for a future reference.

The prediction unit 660 may generate a prediction signal using thesignal stored in the buffer 650. In this case, the present inventionrelates to predict an area within a target image efficiently using anarea within an anchor image. Herein, the anchor image may mean areference image, a reference picture or a reference frame. Theefficiency may be determined by evaluating a mean square error thatquantifies a rate-distortion cost or a distortion within the predictionerror.

The present invention proposes a method for distinguishing a vertex andan edge within a graph and encoding or decoding a residual signal. Forexample, according to the embodiments of the present invention, variousembodiments may be performed through the graph-based transform unit 610.The graph-based transform unit 610 may be included in the encoder 600 orthe decoder 700.

FIG. 7 illustrates a schematic block diagram of a decoder that processesa graph-based signal as an embodiment to which the present invention isapplied.

A decoder 700 shown in FIG. 7 receives a signal outputted from theencoder 600.

An entropy decoding unit 710 performs entropy decoding for a receivedsignal. An inverse quantization unit 720 obtains a transformationcoefficient from the entropy-decoded signal based on the information ofa quantization step size.

An inverse transform unit 730 obtains a prediction error by performinginverse transformation for a transformation coefficient. In this case,the inverse transformation may mean an inverse transformation for thegraph-based transformation obtained from the encoder 600.

A restored signal is generated by adding the obtained prediction errorto the prediction signal outputted from a prediction unit 750, which isperformed in a restoration unit (not shown).

A buffer 740 stores the restored signal for a future reference of theprediction unit 750.

The prediction unit 750 generates a prediction signal based on thesignal stored in the buffer 740 which is restored previously and theprediction vector to which the present invention is applied.

In the present invention, the graph-based transformation obtained basedon a graph parameter may be used in the encoder 600 or the decoder 700.

FIG. 8 illustrates an inner block diagram of a graph-based transformunit as an embodiment to which the present invention is applied.

Referring to FIG. 8, the graph-based transform unit 610 may include agraph parameter determining unit 611, a graph signal generating unit613, a penalty function generating unit 615, a transform matrixcalculating unit 617 and a transform performing unit 619.

The graph parameter determining unit 611 may extract a graph parameterwithin a graph that corresponds to a target unit of a video signal or aresidual signal. For example, the graph parameter may include at leastone of a vertex parameter and an edge parameter. The vertex parametermay include at least one of a vertex position and a vertex number, andthe edge parameter may include at least one of an edge weighting valueand an edge weighting value number. In addition, the graph parameter maybe defined as a set of a predetermined number.

According to an embodiment of the present invention, the graph parameterextracted from the graph parameter determining unit 611 may berepresented as a generalized form. For example, ‘A’ numbers of vertexparameter set may be defined as represented in Equation 18 above. Inthis case, ‘A’ numbers of vertex parameter set may be represented asV-dimensional vectors. And, ‘B’ numbers of edge parameter set may bedefined as represented in Equation 19 above, and this may be representedas V×V matrix.

The graph signal generating unit 613 may generate a graph signal basedon the graph parameter extracted from the graph parameter determiningunit 611. In this case, the graph signal may be defined as Equation 20above.

The graph-based transform unit 610 may define a penalty function set inorder to solve the constraint optimization problem. And accordingly, thepenalty function generating unit 615 may generate a penalty function forcalculating an optimal transform matrix. For example, the penaltyfunction generating unit 615 may define a penalty function set asrepresented in Equation 21 above.

According to an embodiment of the present invention, the penaltyfunctions may be defined to be calculated only for a vertex value. And,the parameters in the same dimension may be calculated using the penaltyfunction as represented in Equation 25 above, and for an edge valuedifference, the penalty function as represented in Equation 26 above maybe used.

The transform matrix calculating unit 617 may generate an optimizationfunction based on at least one of the generalized graph parameter andthe penalty function, and may calculate an optimized transform matrixthat satisfies the optimization function. For example, the transformmatrix calculating unit 617 may derive an optimization function asrepresented in Equation 27 above, based on Equation 25 and Equation 26above.

According to an embodiment of the present invention, by using the graphsignal based on the generalized graph parameter and the penaltyfunction, an optimization function for obtaining an optimized transformmatrix may be generated. For example, the optimization function may bedefined as Equation 24 above.

According to an embodiment of the present invention, an optimizationfunction may be defined based on a penalty function for at least one ofthe penalty function for a vertex and an edge value difference.

The transform performing unit 619 may perform transform using theoptimized transform matrix obtained from the transform matrixcalculating unit 617.

In relation to FIG. 8 above, in the present specification, the procedureof performing the graph-based transform will be described by subdividingit for each function, but the present invention is not limited thereto.For example, the graph-based transform unit 610 may include a graphsignal generating unit and a transform unit, largely. In this case, thefunction of the graph parameter determining unit 611 may be performed inthe graph signal generating unit, and the functions of the penaltyfunction generating unit 615, the transform matrix calculating unit 617and the transform performing unit 619 may be performed in the transformunit. In addition, the function of the transform unit may be dividedinto a transform matrix calculating unit and a transform performingunit.

FIG. 9 is a flowchart for describing a procedure of calculating anoptimized transform matrix based on a generalized graph parameter and apenalty function, as an embodiment to which the present invention isapplied.

An encoder may generate a prediction signal from a received videosignal, and may generate a residual signal by subtracting the predictionsignal from the video signal. A transform is performed for the residualsignal. In this case, a graph-based transform may be performed byapplying the graph-based signal processing technique.

The encoder may extract a graph parameter in a graph that corresponds tothe video signal or a target unit (e.g., transform unit) of the residualsignal (step, S910). For example, the graph parameter may include atleast one of a vertex parameter set and an edge parameter set. Herein,the graph parameter may be represented as a generalized form. Forexample, ‘A’ numbers of vertex parameter set may be represented asV-dimensional vectors as represented in Equation 18 above. And, ‘B’numbers of edge parameter set may be represented as V×V matrix asrepresented in Equation 19 above. This may be performed in thegraph-based transform unit 610, particularly, in the graph parameterdetermining unit 611.

The encoder may generate a generalized graph signal based on thegeneralized graph parameter (step, S920). This may be performed in thegraph-based transform unit 610, particularly, in the graph signalgenerating unit 613.

Meanwhile, the encoder may generate at least one of a penalty functionand a constraint function in order to solve the constraint optimizationproblem (step, S930). For example, based on the generalized graphparameter, at least one of the penalty function set and the constraintfunction set. This may be performed in the graph-based transform unit610, particularly, in the penalty function generating unit 615. Herein,the penalty function may be generated based on the generalized graphsignal.

And, the penalty function may include a first penalty function componentfor a vertex parameter set and a second penalty function component foran edge parameter set. In this case, the optimization function may becomprised of the summation of the first penalty function component forthe vertex parameter set and the second penalty function component forthe edge parameter set. Herein, the vertex parameter set may berepresented as V-dimensional vector, and the edge parameter set may berepresented as V×V matrix.

The encoder may generate an optimization function based on at least oneof the penalty function and the constraint function and the generalizedgraph signal (step, S940), and may obtain an optimal transform matrix(or optimal transform kernel) that satisfies the optimization function(step, S950). This may be performed in the graph-based transform unit610, particularly, in transform matrix calculating unit 617.

Based on the optimized transform matrix which is calculated, a transformfor the target unit may be performed (step, S960).

FIG. 10 is a flowchart for describing a procedure of obtaining agraph-based transform kernel using a generalized parameter set, as anembodiment to which the present invention is applied.

A decoder, to which the present invention is applied, may receive ageneralized graph signal including a graph parameter set (step, S1010).Herein, the graph parameter set may include at least one of a vertexparameter set represented as V-dimensional vector and an edge parameterset represented as V×V matrix. The graph parameter set may betransmitted to a syntax element or may be induced from other informationin the decoder.

The decoder may obtain a graph-based transform kernel of a transformunit based on the graph parameter set and the predefined penaltyfunction (step, S1020). Herein, the predefined penalty function may bethat of generated based on the generalized graph signal. And, thegraph-based transform kernel may be calculated by using an optimizationfunction, and the optimization function is based on the graph parameterset and the predefined penalty function.

In addition, the decoder may decode the transform unit using theobtained graph-based transform kernel (step, S1030). In this case, thegraph-based transform kernel may be calculated by the optimizationfunction based on the graph parameter set and the predefined penaltyfunction. In addition, the optimization function may be comprised of thesummation of a first penalty function component for the vertex parameterset and a second penalty function component for the edge parameter set.The graph-based transform kernel may indicate a value in which theoptimization function is a minimum.

As such, by providing a new method for calculating a graph-basedtransform using generalization of a graph parameter, the presentinvention may avoid the sharp discontinuity problem of vectors fordefining the graph-based transform, may better control the transformproperties, and may be applicable to various applications. Furthermore,an excessive overhead of a bit rate required to encode a graph signalmay be avoided.

As described above, the embodiments explained in the present inventionmay be implemented and performed on a processor, a microprocessor, acontroller or a chip. For example, functional modules explained in FIG.1, FIG. 2, FIG. 6, FIG. 7 and FIG. 8 may be implemented and performed ona computer, a processor, a microprocessor, a controller or a chip.

As described above, the decoder and the encoder to which the presentinvention is applied may be included in a multimedia broadcastingtransmission/reception apparatus, a mobile communication terminal, ahome cinema video apparatus, a digital cinema video apparatus, asurveillance camera, a video chatting apparatus, a real-timecommunication apparatus, such as video communication, a mobile streamingapparatus, a storage medium, a camcorder, a VoD service providingapparatus, an Internet streaming service providing apparatus, athree-dimensional 3D video apparatus, a teleconference video apparatus,and a medical video apparatus and may be used to code video signals anddata signals.

Furthermore, the decoding/encoding method to which the present inventionis applied may be produced in the form of a program that is to beexecuted by a computer and may be stored in a computer-readablerecording medium. Multimedia data having a data structure according tothe present invention may also be stored in computer-readable recordingmedia. The computer-readable recording media include all types ofstorage devices in which data readable by a computer system is stored.The computer-readable recording media may include a BD, a USB, ROM, RAM,CD-ROM, a magnetic tape, a floppy disk, and an optical data storagedevice, for example. Furthermore, the computer-readable recording mediaincludes media implemented in the form of carrier waves, e.g.,transmission through the Internet. Furthermore, a bit stream generatedby the encoding method may be stored in a computer-readable recordingmedium or may be transmitted over wired/wireless communication networks.

INDUSTRIAL APPLICABILITY

The exemplary embodiments of the present invention have been disclosedfor illustrative purposes, and those skilled in the art may improve,change, replace, or add various other embodiments within the technicalspirit and scope of the present invention disclosed in the attachedclaims.

The invention claimed is:
 1. A method for decoding a video signal usinga graph-based transform, comprising: receiving a generalized graphsignal including a graph parameter set, wherein the graph parameter setincludes a vertex parameter set represented as a V-dimensional vectorand an edge parameter set represented as a V×V matrix; obtaining agraph-based transform kernel of a transform unit based on the graphparameter set and a predetermined penalty function; and decoding thetransform unit using the graph-based transform kernel, wherein thegraph-based transform kernel is calculated by an optimization functionbased on the graph parameter set and the predetermined penalty function,wherein the optimization function comprises a summation of a firstpenalty function component for a vertex parameter set and a secondpenalty function component for an edge parameter set, and wherein thegraph-based transform kernel indicates a value that minimizes theoptimization function.
 2. The method of claim 1, wherein thepredetermined penalty function is generated based on the generalizedgraph signal.
 3. A method for performing a graph-based transform basedon a generalized graph signal, comprising: determining a graph parameterincluding a vertex parameter set and an edge parameter set, wherein thegraph parameter set includes a vertex parameter set represented as aV-dimensional vector and an edge parameter set represented as a V×Vmatrix; generating a generalized graph signal based on the graphparameter; generating a penalty function and a constraint function basedon the graph parameter; generating an optimization function based on atleast one of the penalty function and the constraint function and thegeneralized graph signal, wherein the optimization function comprises asummation of a first penalty function component for a vertex parameterset and a second penalty function component for an edge parameter set;obtaining an optimal graph-based transform kernel in which theoptimization function is a minimum; and performing a transform for atransform unit using the optimal graph-based transform kernel.
 4. Themethod of claim 3, wherein the penalty function is generated based onthe generalized graph signal.
 5. An apparatus for decoding a videosignal using a graph-based transform, comprising: a storage deviceconfigured to store the video signal; and at least one processorconfigured to: receive a generalized graph signal including a graphparameter set, wherein the graph parameter set includes a vertexparameter set represented as V-dimensional vector and an edge parameterset represented as V×V matrix; and obtain a graph-based transform kernelof a transform unit based on the graph parameter set and a predeterminedpenalty function; and decode the transform unit using the graph-basedtransform kernel, wherein the graph-based transform kernel is calculatedby an optimization function based on the graph parameter set and thepredetermined penalty function, wherein the optimization functioncomprises a summation of a first penalty function component for a vertexparameter set and a second penalty function component for an edgeparameter set, and wherein the graph-based transform kernel indicates avalue that minimizes the optimization function.
 6. The apparatus ofclaim 5, wherein the predetermined penalty function is generated basedon the generalized graph signal.
 7. An apparatus for performing agraph-based transform using a generalized graph signal, comprising: astorage device configured to store video data; and at least oneprocessor configured to: determine a graph parameter including a vertexparameter set and an edge parameter set, and to generate a generalizedgraph signal based on the graph parameter, wherein the graph parameterset includes a vertex parameter set represented as a V-dimensionalvector and an edge parameter set represented as a V×V matrix; generate apenalty function and a constraint function based on the graph parameter,to generate an optimization function based on at least one of thepenalty function and the constraint function and the generalized graphsignal; and obtain an optimal graph-based transform kernel in which theoptimization function is a minimum, wherein the optimization functioncomprises a summation of a first penalty function component for a vertexparameter set and a second penalty function component for an edgeparameter set; and perform a transform for a transform unit using theoptimal graph-based transform kernel.
 8. The apparatus of claim 7,wherein the penalty function is generated based on the generalized graphsignal.